# Math tips

## Terms

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if you are given several groups of different sizes and with diff mean scores, how would you go about finding the overall (arithmetic) mean score for those groups?
(mean score 1)(class size 1) +(means score 2)class size 2)/ (class size 1+class size 2)
In a certain yr, Minnesota produced 2/3 and MI produced 1/6 of all the iron ore produced in the US. From this info, how much did all the other states produce?
1/6 + 2/3 = 5/6

6/6-5/6 = 1/6 produced by all other states
Minnesota produces 2/3 of the iron ore, Michigan produces 1/6 and all of the other states combined produce 1/6. If all other states produce 18 mill tons, how many tons does Minnesota produce?
1/6 (total production)= 18
total production = 108

2/3 (108) = 72 produced by minesota
Is 0 an integer?
If so, is it a pos integer or a neg one?
0 is an integer, but it is neither a pos or a neg one
5000 - 4879 = ??
121
~Maintain 9s no matter how many times you carry.
To express a number as a percent, what should you do to it?
multiply by 100
To express a percent as a number, what should you do to it?
divide by 100
How do you determine % increase?
1. find the difference between the before and after salaries.

2. diff/orig # = decimal

3. decimal *100 = %
A number is divisible by 3 if...
the sum of all of the digits is divisible by 3
A number is divisible by 4 if...
the number formed by the rightmost 2 digits is divisible by 4
A number is divisible by 6 if...
it is divisible by 2 & 3
A number is divisible by 9 if...
the sum of all of the digits is divisible by 9
10(sq rt of 3)/2(sq rt of 3)=??
5
If the ratio of men to women on a comittee of 20 members is 3 to 2, how many members of the committee are women..What's the key to solving this?
3men + 2 women = 5 total

2 women/5total = xwomen/20 total
What should you do with neg exponents (ie X^(-a))?
bring them to the denom
1/(x^(a))
when multipling two numbers that have the same bases, what should you do with the exponents?
When dividing 2 numbers with the same bases, what do you do with the exponents?
subtract them
(x / y ) ^(a) = ??
x^(a)/ y^(a)
(x^(a))^(b)
x^(a*b)
.0023 is the same as
2.3 *10^(-3)
Move decimal rt, then exponent is negative. (Move it left, exponent is pos)
5 * 10 ^(-2) is the same as
.05
When multiplying by a # that has a neg exponent, move the decimal pt left.
Adding or subtracting a constant to an inequality does what to the direction of the inequality?
nothing
Multiplying or dividing an inequality by a constant does what to the direction of the inequality?
direction is preserved if constant is pos, switched if the constant is negative.
In a driving competition, Jeff and Dennis drove the same course at average speeds of 51 mi/hr and 54 mi/hr, respectively. It took Jeff 40 min to drive the course. How would you represent the distance traveled by Jeff?
d = r (t)
d= 51 mi/1 hr *(1hr/60min)*(40min/1)
d= 34
In a driving competition, Jeff and Dennis drove the same course at average speeds of 51 mi/hr and 54 mi/hr, respectively. It took Jeff 40 min to drive the course. How would you represent the distance traveled by denis?
d = rt
d = (54 mi/1 hr) * (1hr/60 min) (x min/1)
In a driving competition, Jeff and Dennis drove the same course at average speeds of 51 mi/hr and 54 mi/hr, respectively. It took Jeff 40 min to drive the course, how long did it take dennis?
***the distances are the same, so set the eq equal to one another
(51)(40/60) = 54 (x/60)
sum =
product =
product = multiply together
I think that 4>x is the same as...
x<4
In 82, what is the units digit and what is the tens digit?
2 is the units digit, 8 is the tens digit
**Pat invested a total of 3,000. Part of the money yields 10 percent interest per year, and the rest yields 8 percent interest per year. If the total yearly interest from this investment is \$256, how much did Pat invest at 10 percent and how much at 8 pe
Well, we know pat invests two sums of money. let's call them x and y. We know that x + y = 3000, because he invests 3000. part of it gives 10 yield, and the other gives 8. total interest is 256. This means that 10%x + 8%y = 256, if we let x be the 10 yield part, and y be the 8 yield part. Now, we have two equations and two unknowns...

eq1 --> x + y = 3000
eq2 --> .1x + .08y = 256

Let's start by multiplying eq2 by (-10) to get

neweq2 --> -x - .8y = -2560

Now, we will ad neweq2 and eq1 to get

.2y = 440

And that leaves us with y = 2200.

If y = 2200, then

x + 2200 = 3000, or

x = 800

So, the 10% yield was 800, and the 8%yield was 2200

Kathleen's weekly salary was increased 8 percent to \$237.60. What was her weekly salary before the increase?
original salary) + 8% of (original salary) equals 237.60

lets call her origianl salary x. This gives us an equation

x + 8% x = 237.60

8% is the same as .08, so

x + .08x = 237.60, and we add 1 and .08 to get

1.08x = 237.60

Now, we divide 237.60 by 1.08 to get the answer...it is

220.00

Hope this helps

Two cars started from the same point and traveled on a straight course in opposite directions for exactly 2 hours, at which time they weree 208 miles apart. If one car traveled, on average, 8 miles per hour faster than the other car, what was the average
s = d/t

s = 208/2
s = 104

x + (x + 8) = 104

x + x + 8 = 104

2x + 8 = 104

2x = 96

x = 48

x + 8 = 48 + 8 = 56

the answer is 48 mph and 56mph

---------------------

Since Speed = distance/time, i just did it like this

x + (x + 8) = 208/2

and i got

x + x + 8 = 104

and you know the rest.

average speed of the 2 cars = (average distance)/(average time)
**A group can charter a particular aircraft at a fixed total cost. If 36 people charter the aircraft rather than 40 people, then the cost per person is greater by \$12. What is the cost per person if 40 people charter the aircraft?**
Let's assign this fixed cost as C. For 36 people, the price per person in simply C/36 while we have C/40 per person for the 40 people group.
But the question says that the difference in the cost per person for the two groups is \$12. Therefore,
C/36 - C/40 = 12
multiplying boh sides by 360, we have
(360)C/36 - (360)C/40 = (360)12
10C - 9C = 4320
C = \$4320
But the question requires the cost per person for the 40 people group
= C/40
= 4320/40
= \$108
Three years ago men made up two out of every three internet users in America. Today the ratio of male to female users is about 1 to 1. In that time the number of American females using the internet has grown by 30,000,000, while the number of males who u
Let x = original number of female users.
So the total internet-user population three years ago was 1x + 2x = 3x.
The male population has doubled, (2x + 2x = 4x)
But the female population now equals the male one, so 1x + 30,000,000 = 4x. Therefore x = 10,000,000.
So the total internet-user population three years ago was 30,000,000. How much more than that is it now, if x=10,000,000 and the total number of users now stands at 8x (4x + 4x)?
This means that three years ago the internet user population was 30,000,000, while now it is 80,000,000 (4x + 4x). So it has increased by 50,000,000.
Column B can be re-written as a^1/4*a^1/2. Because Column A is a^1/2, when a < 1 then????
when a<1 a ^1/2 is bigger, when a>1, a ^1/8 is bigger.
???????
Two years ago, a rabbit breeder had 75 rabbits. Since then 500 rabbits have been born, 220 have died and 185 have been sold. How many rabbits does the breeder have now expressed as a percentage of the number she had two years ago?
227%
How do you add up 6, 7 and sq rt of 85?
treat them seperately.
13 + sq rt 85
to find the x intercept, you set what term equal to zero and solve?
y
The sum of the measures of the interior angles of an n sided polygon is....
(n-2)(180)

Note: triangles, quands, pentagons are all ex of polygons
when finding the area of a parallelogram...
just take the area of the length times the height of the straight line inside the figure (not the two slanted lines on the outside).
How do you find the arc length of a circle?
length of arc/ Circumfrence = degrees of section/ 360
Finding the area of a piece of a circle...
(area of the piece/area of entire circle) = (degrees of the piece/ 360)
how do you find the surface area of a rectangular solid?
2 (lh + wl + wh)
2 times all possible combos of the sides
After spending 5/12 of his salary, a man has 140 left. What is his salary
1-5/12 = 7/12
(7/12) = 140/x
The greenpoint factory produced 2/5 of the consolidated brick company's bricks in 1991. If the Greenpoint factory produced 1,400 tons of bricks in 1991, what was the consolidated brick company's total output that yr?
If 2/5 outpoot was 1400, then 1/5 is 700. 700*5=3500
Dividing by a number is the same as ....
multiplying by its reciprocal
If the ratio of boys to girls in a class is 5 to 3, which of the following could not be the number of students in the class?
a) 24
b) 40
c) 96
d) 150
e) 224
5x + 3x = 8x
The number of students must be a multiple of 8
If you recieve something at 25 percent off of the original price, what are you actually paying for it?
You're paying 75 % of the original price.
selling price =
selling price= cost to store + profit

For a profit of 25%,
selling price = original + (original)(.25)
If an item costs \$800 after a 20 % discount, 800 is equal to what percent of the original price?
The discounted price is equal to 80 percent of the original price.

original * .80 = cost after discount which in this case is 800
If carol can do a job in five hours, what portion of the job can she do in 1 hr?
1/5 job in 1 hr.
If Carol can do 1/5 of job in 1 hr and Steve can do 1/10 of the same job in an hr, how much of the job can they do in one hour if they are working together ??
1/5 + 1/10 = 3/10 of job
If a person can do 3/10 of a job in an hour, then then how long will it take them to do 1/10 of the job? The whole job?
(1/10job / x hr) = (3/10job / 1 hr)
x=1/3 hr for 1/10 of job.

Whole job is 10/10
(10/10 of job/ x hr) = (1/10 of job/ 1/3 hr)
the ratio of sides in a 30-60-90 triangle is...
1/ sq rt 3 /2
formula for the slope of a line
y1-y2/ x1-x2
If enrollment was 17% higher for freshmen at City College in the Fall 2000 semester than the Fall 1999 semester, approximately what was the enrollment of freshman in the Fall 1999 semester?
Note:
% of fresh= 40%, total enrollment in 200 is 105
We need to be careful here, the temptation is to just take 40% - 17% = 23% of the enrollment of 10,500. However, keep in mind that we are talking about 17% higher in terms of enrollment, not compared to the other classifications and enrollment may or may not have been the same overall for 1999 and 2000.

The first thing we need to find is the actual enrollment for freshmen in 2000. According to the chart, freshmen made up 40% of the 10,500 enrolled. Taking 40% of 10,500 we get:

4200 is 17% higher than the enrollment of freshmen in 1999. Letting x represent the 1999 freshmen enrollment and putting this into an equation we get:

*17% higher means we ADD 17% (or .17) of the amount

*Divide BOTH sides by 1.1

4200= x+ .17x
If account codes for a certain company are assigned as follows: two letters and then three one digit numbers, how many different account codes can be made? Note that letters and digits can be repeated.
We can use the fundamental counting principle to answer this question. Basically we need to take the product of the number of ways each event can occur.

There are 5 stages or events: letter 1, letter 2, digit 1, digit 2, and digit 3.

In general, there are 26 letters in the alphabet. Neither letter has any restriction, so there are 26 possibilities for each letter.

In general there are 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. There are no restriction on any of the digits, so each one of those has 10 possibilities.

Putting that all together we get:

Letter 1 Letter 2 Digit 1 Digit 2 Digit 3
Total
26 x 26 x 10 x 10 x 10 = 676000

There are 676,000 different account codes possible
Dividing by a number is the same as.....
multiplying by its reciprocal
Being a conscientious driver, Suzy stayed at or below the speed limit while traveling down the interstate. Overall, she went an average rate of 65 mph and it took her 10 hours to complete her journey. If she traveled for 6 of her 10 hours at 70 mph, what
d=rt
I solved d/t =r

(6*70) +4(x) / (10) = 65
area
for triangle vs. rectangle
area of triangle = 1/2 bh
area of rectangle = l * w
volume of cylinder
The volume of a cylindrical tank is pi r sq h
Parallelogram area can be found by taking b*h
Two types of parallelograms are rectangles and squares. What are the modified formulas for recs and sq?
Area of sq= (side)^(2)
area of rec = l*w
formulas for surface area & volume of a rectangle
**surface area = 2lw + 2lh + 2 wh

Volume = (l*W*H)
circle area and circumference
a = pi*( r ^(2))
C= 2 pi *r
Surface Area and Volume of a Right Circular Cylinder
Surface Area: 2 pi rh +2 pi r^(2)

Volume:
2 pi (r ^2) h
If 25 students in one class had an average of 93% and 20 students from another class had an average of 98%, approximately what is the average in percent of all 45 students?
In general, when finding the average of tests, you add the tests together and then divide by the number of tests. On this problem we have two separate averages to start with. So we need to find the total number of points we have using that information and then divide by the number of students, which in this case is 45.

If 25 students averaged 93, then the total number of points would be (25)(93) = 2325.

If 20 students averaged 98, then the total number of points would be (20)(98) = 1960.

So the total number of points is 2325 + 1960 = 4285.

To find the average, take the number of points and divide by the number of students:
4285/45= 92.5
The overall average percent is 95.2%.
If enrollment was 17% higher for freshmen at City College in the Fall 2000 semester than in 1999, how many stud in 1999? How would you represent this?
number of fresh in 200 = x + .17x

where x = number of fresh in 1999
what is (4 (sq rt of 2)^(2)) ?
what is (4 (sq rt of 2)^(2))
16*2=32
amount of the degrees on the interior of a figure=
(n-2)*(180)
What is the relationship between radius and diameter of a circle?
2 * r = d
What is the formula for the arc length of a circle?
arc length = (x/360)* 2 pi r
What is the formula for the area of a sector?
area of sector= (x/360) * pi r ^(2)
sq rt 3 + sq rt 3 =
sq rt 3 + sq rt 3 = 2 sq rt 3
sq rt 3 * sq rt 3 =
sq rt 3 * sq rt 3 = 3
volume of a sphere
V = (pi) *(r sq)*h
When taking (0.4 * 0.00065), how many places over would you move the decimal?
5 places for .00065 plus 1 place for 0.4.
(9/4) ^(-1)
1/ (9/4)=
4/9
If the equation 4x2 + 6x + c = 0 has exactly one root, what is the value of c?

9/4
3/4
0
-3/4
-9/4
You can solve this a couple of ways. The easiest is to know the general solution for the roots to a quadratic equation ax2 + bx + c = 0 is:

x = minus b + or minus the sq rt of b sq minus 4ac all divided by 2a
If there is to be one unique root, then the value of the square root part has to be zero, so that the + or - part is just + or - 0. So we have that b2 = 4ac. And since a = 4, and b = 6, we automatically get 36 = 16c, or c = 9/4.

Alternatively, if you didn't know the formula for the roots of the quadratic equation, you could note that in order to get 4x2 you would have to have either of the following two factorizations:
(4x + s)(x + t) or (2x + w)(2x + v)
Solving each of these with the constraint that you get equal roots (i.e. s/4 = t) and that the b term in the quadratic is 6 (i.e. 4t + s = 6) yields st = wv = 9/4.
a + b = 3c

a - b = c

If you are also told that b - c = 0, how many unique solutions are there for b?

0
1
2
3
Infinitely many values satisfy b.
There are three unknowns and three equations, which means there will be a unique solution provided all the equations give distinct information. Although it is not immediately obvious that the last equation b - c = 0 is redundant, you can see it by combining the first two equations. Subtracting the second equation from the first gives you 2b = 2c, which is equivalent to saying that b - c = 0. So you really only have two equations and three unknowns, and therefore there are an infinite number of solutions
When Larry drives the 12 miles he must travel to get to work, it usually takes him 30 minutes. On one day he travelled the first 6 miles at half his usual average speed. How fast must he drive for the next 6 miles in order for him to get to work at the u
To answer this question you must keep track of the total trip time. Larry's trip usually takes him 30 minutes. If he travels 6 miles at half his usual speed, then he has already driven for 30 minutes. No matter how quickly he drives the remaining 6 miles, his total trip time will be more than 30 minutes.
Can integers be negative?
yes
Is zero an integer?
yes- it is an even integer.
Integers are whole numbers

note that Zero is not pos or neg.
Given that a, b, and c are consecutive integers, which of the following must be even?
I. a + b + c
II. abc
III. b(a+c)

I only
II only
I and II
II and III
I, II and III
There are two possibilities: either a and c are even and b is odd, or a and c are odd and b is even.

In II, we multiply the three together. Since at least one of them has 2 as a factor, the product must have 2 as a factor, so it's even.

In III, we need to look at either case. If b is even, then the product is even for the same reason as above. If b is odd, then the sum (a+c) is even, and once again the product of b(a+c) is even
Two different coins are tossed.
The number of possible outcomes
of tossing the two coins.
We can use the fundamental counting principle to answer this question. Basically we need to take the product of the number of ways each event can occur.

There are 2 stages or events: coin 1 and coin 2

There are two possibilities for each coin, heads or tails.

stages * probabilities
The number of ways to arrange
five CDâ€™s on a shelf.
P = n! (n-r)!

Where n is the number of CD's we have to choose from and r is the number of CDâ€™s we are using at a time
0! = ??
O! = 1
fundamental counting principle determines ____, while the permutations formula determines ____
fund counting principle determines the number of outcomes, permutations formula (ie n!) determines how to arrange something in order.
to determine the number of outcomes, take...
# of stages or events * choices for each question
taking a fraction to an exponent of 3 will make it ____ than taking the same fraction to an exponent of 2 (or smaller).
larger. The more times that you multiply a fraction by itself, the smaller it becomes.
(x - 2)(y - 1) = 0
All the equation tells us is that either x=2 or y=1.

But if x=2, then y could be anything. And if y=1, then x could be anything
definition of unique solution
solution is unique if no other set of coordinates is also a solution. (x 2 =1 does not have a unique solution, since both x=1 and x=-1 are valid solutions.)
3x + 4y = z

3x + 4y = -z

Under which condition will there be a unique, real-valued solution for x, y, and z?

z = 0
z = 4
z = 3
6x = -8y
None of the above conditions guarantees a u
This one takes some thought. For there to be any real-valued solution for x and y, then we must have z = 0. But if z = 0, then we only know that 3x + 4y = 0, for which there is no unique solution. So in the end, none of the conditions guarantees a unique solution for x, y, and z.
greatest growth rate and greatest increase in growth rate are two different things. How?
on a height gain vs. yrs of age graph, highest pt on height gain indicates greatest growth rate and the largest positive upward slope indicates the greatest increase in growth rate.
Of the following cities, which had the greatest percentage decrease in the number of burglaries from 1998 to 1999?

Chicago
Houston
New York
Phoenix
Providence
When taking the test, you should do as few tedious calculations as possible. It is possible to get the answer to this question by methodically dividing the 1999 number of burglaries by the 1998 number for each city. But a shorter way is to eyeball the fractions of the 1999 number over the 1998 number, and judge for which the city the fraction is smallest. The city where the ratio of 1999 to 1998 burglaries is smallest will be the city that has the largest percent reduction in burglaries. The two most plausible cases are Chicago and Houston.
For Chicago the ratio is roughly 30 / 36 or, simplified, 5 / 6 .
For Houston the ratio is roughly 60 / 76 or, simplified, roughly 4 / 5 .
Therefore the percent reduction in Houston is about 20 % compared with something close to 16.6 % in Chicago.

The greatest percent decrease in burglaries will come from the city where the ratio of buglaries from the two yrs is the smallest.
Which is bigger, sq rt 3 + sq rt 3 or sq rt of 6
sq rt of 3 + sq rt of 3. (This can be written as the aq rt of 12, which is larger than the sq rt of 6)
sq rt of 3 + sq rt of 3 =

sq rt 6 div by sq rt of 10=
sq rt of 3 + sq rt of 3 = 2(sq rt of 3)

sq rt 6 div by sq rt of 10= sq rt (3/5)
If you are being asked about the perimenter of a figure of which a circle is a part, what do you need to find out about the circle.
For perimeter, use the circle's circumference
When using pythag theorem, you can NOT reduce the sides of the triangle when finding the third side unless you bring everything back up to scale after you've solved.
When using pythag theorem, you can NOT reduce the sides of the triangle when finding the third side unless you bring everything back up to scale after you've solved.
in 5-2(4) which do you do first:
distribute the 2
OR
take five minus 2 and then distribute?
distribute the two:
5-8
What is 70 + 75+ 55+ 2x?
200 + 2x
The volume of a sponge decreased by 12 percent as it dried. If the sponge's volume was 308 cubic inches after it dried, what was its volume before it dried?
V-.12V = 308
0.88V = 308

Ans: 350
formula for the volume of a cylinder
area of a circle * h
How would you get 4 ^ (5) factored into a pwr of 2?
2^(2) = 4, so

(2^(2))^(5) = 2^(10)

Note: (2^(2))^(5) is not the same as 2^(2) * 2^(5). 2^(2) * 2^(5) = 2^(7)
How would you factor 2 ^(21)?
2^(8) * 2^(13)

Same base, add the exponents. If a number with exp is enclosed in parenth, and followed by an exponent, MULTIPLY the two exponents together?
2^(13)/ 2^(8) + 2^(10) =
2^(5) / 1+ 2^(2)

Subtract the largest 2^# that will fit into all of the terms..in this case, use 2^(8)
if you take a whole number to an exponent, you get a larger number, but if you take a fraction to a exponent...
you get a smaller number
5^(x)/ 5^(x-1) =
5^(x)/ 5^(x-1) = 5^(x -(x-1))
5 ^(x-x+1)
5^(1) = 5
x, y, and z are all angles in a triangle. If x = 2z+1, which angle is bigger, x or z?
x is bigger.
(You had to add all kinds of stuff to z to make it equal to x, which means that z must have been smaller to begin with).
How many multiples of 28 less than 300 are also multiples of 16.
Write out the prime factorization for 28 and for 16. Find which factors they have in common and be sure not to use any of them more than once in the final multiplication (ex. 2*2 from 1 number and 2*2*2 would just be 2*2*2) . Multiply together the numbers that remain.
In this example:
28: 2*2*7
16: 2*2*2*2
so
2*2*2*2*7 = 112
112 is the smallest positive multiple of the two numbers
On comparison questions, always try to determine if x could be equal to multiple values...if so you will use ans can not be determined..
check for fractions,etc.
See if the problem specifies that x must be pos, for example.
What is the formula for percent incr?
Percent incr = difference/original *100
If 30 students in one class had an average of 80% and 10 students from another class had another average of 90%, what is the average in % of all 40 students?
In general, when finding the average of tests, you add the tests together and then divide by the number of tests.

On this problem we have two separate averages to start with. So we need to find the total number of points we have using that information and then divide by the number of students, which, in this case, is 40.

If 30 students averaged 80, then the total number of points would be (30)(80) = 2400.

If 10 students averaged 90, then the total number of points would be (10)(90) = 900.

So the total number of points is 2400 + 900 = 3300.

To find the average, take the number of points and divide by the number of students:

The overall average percent is 82.5%.
when you multiply both sides of an inequality by a negative number, you must...
switch the direction of the inequality
The distance/rate formula is...
distance = rate * time
Overall, Carly went an average rate of 50 mph and it took her 8 hours to complete her journey. If she traveled for 6 hours at 55 mph, what constant speed did she go for the remaining 2 hours?
20 - 3. Overall, Carly went an average rate of 50 mph and it took her 8 hours to complete her journey. If she traveled for 6 hours at 55 mph, what constant speed did she go for the remaining 2 hours?

The distance/rate formula is .

This problem has several parts to it.

First let see how many miles Carly traveled overall. Plugging in the rate of 50 mph and the time of 8 hours we get:

Carly traveled a total of 400 miles.

Next we start breaking it down. It says that she traveled for 6 hours at 55 mph. From this we can figure out how many of the 400 miles she traveled at this speed:

She traveled 330 miles at the 55 mph.

How many miles does that leave us for the unknown speed?
400 - 330 = 70

For the last part, we are again going to use the distance/rate formula, but now we are looking for the rate. Plugging in 70 for the distance and 2 for the time we get:

*Inv. of mult. by 2 is divide by 2

Carly traveled 35 mph for 2 hours of her trip.
What is the area of a square that is inscribed in a circle whose area is 25pi square feet?
Ultimately, we need to find the area of the square. That means we need the area of a square formula which is .

Since we were not given the side of the square, it looks like we will have to go out and find it.

Since the area of a circle is found by taking pi times the radius squared, this means the radius of our circle is going to be 5 feet, since its area is square feet.

Let's look at a visual of this:

It looks like the diameter of the circle is the diagonal of the square.
Just a reminder that all four sides of a square are equal and the diagonal of a square is equal to .

Putting 10 in the diagonal we get:

*Inv. of mult. by 2 is divide by 2

*Square root method

The side of the square is feet long.

Putting in into the area formula we get:

The area of the square is 50 square feet.
area of a rectangle
vs
area of a parallelogram
area of rectangle = l*w

area of a parallelogram = b*h
integers
-are whole numbers
-can be positive or negative
-0 is an integer
-FRACTIONS are NOT integers
What is the percent formula?
part/whole *100 = percent
What percent is 10 of 20?
10/20 = 50 percent
10 is 20 percent of what number?
50
volume of a cylinder
pi* (r sq)* h
To determine the sum of averaged numbers...
(number of numbers) * (calculated average)
For smallest/largest number factored by problems..
factor into primes and multiply by different combos
-v = 7u

What is the ratio of u to v?
-v = 7u
-1 = 7 (v/u)
-1/7 = v/u

Do the necessary operations to get the desired ratio of variables on one side, then solve to get the numbers on the other
if there is only one equation with two unknown variables,
then their is no unique solution

example: 3 +xz = 19
15/z +5x = 95/z
One fifth of criminals are hard-core criminals. The hard-core criminals commit two-thirds of the criminal acts.
A B
The ratio of the number of criminal acts committed by the average hard-core criminal to the number committed by the average cr
Well done. The hard-core criminals commit twice as much crime (two thirds is twice as big as one third), with only a quarter as many people (one fifth are hard core, four fifths are not hard core). So that's eight times as much crime per person.
Does a positive number get bigger when you add or when you subtract another positive number?
Which of the following inequalities is implied by y - x > 3?
Add x to each side and subtract 3 to get: y - 3 > x, which when turned around yields x < y - 3

***You don't have to switch the direction of the inequality if you are simply adding or subtracting, but you must switch the direction of the inequality when you:
1. mult by neg number
2. have circumstances like
y-2>x, which is the same thing as x< y-2
What was the average rate of warming in the northern hemisphere from 1900 to 1960?

Where's the decimal point?
This one you can do backwards--look at the answer choices. Multiply one of them by the 6 decades to get the total temperature rise for that choice of answer. Then adjust by factors of ten until you get the right number, 0.24 degrees.
So:
That's 0.24 degrees over 6 decades, so it's 0.24/6=0.04 degrees per decade. Nice work!
What was the increase in CFC concentration in the northern hemisphere from 1900 to 1960?

4.5 parts per million
5 parts per million
5.5 parts per million
6 parts per million
6.5 parts per million
The CFC concentration in 1900 is about 1.25 parts per million (halfway between 1.00 and 1.50). In 1960, it is about 5.75 parts per million. The difference is 4.5 parts per million. Well done
The chart represents revenue and expenses for a company in 1998.
If the company's revenue equals the company's expenses, and if manufacturing expenses totalled \$120,000, how much revenue did the company have from sales?
Excellent. We know that 15% of the total expenses is \$120,000, so the total expenses are \$800,000. We were told that revenue = expenses, so the total revenue from sales is 92% Ã— \$800,000. Final answer, the company took in \$736,000 from sales.

*Need to find an example where it's not just .15x=120,000 but .15x +x. In what kind of problem does that occur?
covert .8% to a decimal
.8/100 = 8/1000=1/125
In terms of straight-line distances, it is 75 miles from Abbotsville to Bartsburg, and 12 miles from Bartsburg to Crawford. Which of the following is a possible straight line distance from Abbotsville to Crawford?

30 miles
50 miles
The triangle inequality states that the largest side of a triangle is less than the sum of the other two sides. The straight lines joining all possible pairs of three towns form a triangle. The missing distance has to be large enough so that 12 + x > 75 and small enough so that 75 + 12 > x . Of the five given numbers, only 70 satifies the conditions
After spending 5/12 of his salary, a man has 140 left. What is his salary?
12/12-5/12 = 7/12
7/12x = 140
x=240
In a local election, votes were cast for 3 diff people in the ratio of 4:3:2. If there were no other candidates and none of the 1800 cast more than one vote, how many votes did the second person receive?
4x+3x +2x =1800
9x=1800
x=200
3(200) = 600
What percent of 1600 is 2?
1/8%

part/whole *100%
(2/1600) *100% = 200/1600% = 1/8%
Are neg numbers considered even (or odd)?
Yes
taking the sq rt of something is the same as taking it to 1/2. this is useful when you take the sq rt of 1 million be/c..
1 *10^(6) ^(1/2) = 10 ^(3) = 1000
Column A:x
Column B:125% of 80% of x
Yup. To take 80% of x is the same as multiplying x by 80/100 or 8/10. To take 125% of x is the same as multiplying x by 125/100 or 5/4. So Box B is

(8/10)*(5/4)*x = 40/40*x = x
When dividing 8x^(3) by 2x^(2), what do you do?
8/2 = 4
x^(3)/x^(2) = x^(3-2) = x^(1)
x^(0) =
1
In total, 100 papers were sold. some were sold at 25 cents and others at 40 cents. How would you represent this algebraeically?
x and 100-x
when backsolving, use ans choice c (middle) first. If c is too big, then you can elminiate c and the two choices after it. If c is too small, you can eliminate c and the two choices before it.
when backsolving, use ans choice c (middle) first. If c is too big, then you can elminiate c and the two choices after it. If c is too small, you can eliminate c and the two choices before it.
6 > x+4 > 4
2>x>0
0<x<2

(4 has to be subtracted from both sides)
10^(3) is equivalent to what number?
10^(3) = 1000
When ten is raised to a power, the final answer will have (in total) the number of zeros specified by the power, (NOT one zero plus however many zeros are specified by the power).
the two diagonals of a rectangle...
are equal
an exterior angle of a triangle is equal to...
the sum of the opposite two interior angles
In a group of 60 workers, the avg salary is 80 per day per worker. If some of the workers earn 75 per day and all the rest earn 100 per day, how many workers earn 75 a day?
(avg salary) (# of workers)=total income

If the average salary of the 60 workers is 80, the total amt recieved by the workers is 60*80, which = 4,800
What is the law of exponents and how do you use it?
Get numbers on both sides of the equal sign to have a common base. Multiply the existant exponent by the common base's exponent on each side, set these two products equal to one another. See pg 305 of Barrons
product vs quotient
product= result of things that have been multiplied together.
quotient = result of things that have been divided by one another
how would you express x% of y?
(x/100)*(y)
when substituting numbers, the best ones to use are..
1,0, -1.
Fractions can also be helpful.(I don't think that fractions are integers, I think integers must be whole numbers)
I always try to test two sets of numbers that are diff from one another (ie, fraction & whole number, pos & neg, etc)
Don't forget the read the restrictions on the numbers (ie, consecutive integers).
Which is larger, 7/6 hr or 6/7 hr?
A larger number in the denominator yields a smaller fraction.
ie, 7/6 > 6/7
When looking at a + slope line from left to rt, it will slope up . When looking at a - sloping line from left to right, it will slope downward
When looking at a + slope line from left to rt, it will slope up . When looking at a - sloping line from left to right, it will slope downward
56<5c<64
How solve?
break this into two separate inequalities.
solve for two separate values
plug numbers back in to be certain that both of them work
How do you express 40% as a fraction?
40/100

take whatever number is in front of the % (even if it is a decimal) and place it over 100.
If (7^(a)) *(7^(b)) = (7^(c)/ 7^(d)), what is d in terms of a, b, and c?
7^(a+b) = 7^(c-d)
a+b = c-d
d=c-a-b
For 12, You'll have a finite set of factors (from -12 to 12, not including zero) but an infite set of multiples (includes 0, as well as numbers beyond 12 that are divisible by 12).
In determining the factors of 12..
Is 12 a factor of 12?
I
yes to all
factors are also known as divisiors, and they are..
finite
the remainder is always less than the divisor
the remainder is always less than the divisor
Factors of 10
1,2,3,5,6,10,15,30

-when determining factors of a number, don't just write out the prime factorization, write out ALL numbers that can be divided into the number
-always include 1 and that number as factors of a number
-zero is an even integer,but it is not pos or neg. In general, though, integers are simply whole numbers that are pos or neg. ZERO is NOT a factor, it is only a multiple
When doing negative number^(exponent)..
be cognizant of whether the exponent is even or odd. If a negative number is taken to a even exponent, the ans will be positive. If a neg number is taken to an odd exponent, the ans will be negative.
105 - 99 =
6

When getting the 15 for the far right, the 1 becomes a zero and the zero(in middle) becomes a 9.

I think that if there had been more zeros in between they would have been nines too.
if n is less than one (ie, it's a fraction), then taking the square root of n will yield a fraction that is ...
larger than the original n
the fraction with the larger denom is the smaller fraction.
For example,
1/4<1/2
squaring a whole number makes it larger, but squaring a fraction makes it ...
smaller
-5 < -3, and

-2/3*5/3 < -2/3*3/5
Explain.
5/3 is a larger fraction than 3/5. When you take 5/3 times a neg number, you'll get a number whose absolute value is larger than the same neg number multiplied by 3/5 BUT neg numbers with a larger abs value are actually smaller than numbers with sm abs value (ie, -5<-3)
What is 6/1000 expressed as a percent?
0.6%

0.6%= 0.6/100=6/1000
Charlie gave 20% of his baseball cards to Kenne and 15% to Paulie. If he still had 520 cards, how many did he have originally..What is this problem really asking
15+20 = 65
520 is 65% of what number?
For any positive number a: a%of 100 is a. So,
91.2% of 100 = 91.2
and
if the population in town a incr by 60 percent, how would you represent this?
60% of 100 = 60
100+60 = 160
***
55% = 55/100

for percent problems, if a poplulation goes down by 30 percent,
***
100 * 30% = 30
100-30 = 70
you are being told that # of cards is a certain percent of what number
In 1995, the population of town B was what percent of the population of town z?
pop for B= (x/100)* pop for Z
and solve for x

what percent of means that you should use (x/100) and solve for x
100*40%= 40--> 140
Do this only for probs in which the pop has increased/decreased
In contrast,
use a soln that is 80% sugar= 80 oz sugar, 80 oz water.
percent formula

% increase formula
percent= (part/whole)*100

% incr= (final-original/original)*100
A club had 3 boys and 5 girls. During a membership drive the same number of boys and girls joined the club. How many members does the club have now if the ratio of boys to girls is 3:4?
(3+x/5+x) = 3/4
x=3

set the new ratio equal to the ratio that represents the number of cards added
when can you do the 2x+5x+8X= 3000?
With ratio probs,when they give you 2:5:8 and a total quantity (3000) that needs to be divided amongst the three groups.
Sally invited the same number of boys and girls to her party. Everyone who was invited came, but 5 additional boys showed up, causing the ratio to be 4:5.
how many girls were at the party?
you can solve for a ratio or a percent, but not a specific number unless you're given some kind of total or a scenario similar to what happens in the following situation:
Sally invited the same number of boys and girls to her party. Everyone who was invited came, but 5 additional boys showed up, causing the ratio to be 4:5.
(x/x+5)=4/5
x=20
On thurs, 20 of 25 students in a chem class took a test and their avg was 80. On Fri, the other 5 took the test and their avg was 90. What was avg for entire class
20*80=1600
5*90=450
1600+450=2050
Class avg: 2050/25=82
What is the avg (arithmetic mean) of the degree measures of all eight angles, which of the following is true?
2^9 + 2^19

You can't ADD two numbers with similarly based exponents. You can MULTIPLY two of the numbers together by adding their exponents or Divide by subtracting the exponents
In the comparison questions, actively look for something that would make them equal. Then, if you can find just one other choice that would make them different, the ans must be D.
Also, 1,-1,0, and fractions are great numbers two try. >1 posibility means that ans must be D
a^2 -b^2 can be represented as
(a+b)*(a-b)
formula for the interior angles of a figure (or at least I think so...)
(n-2)180
Make sure you read the question carefully and solve only for what the question is asking! (eg, you can substitute something in for a, but the problem is actually asking something about b)
Make sure you read the question carefully and solve only for what the question is asking! (eg, you can substitute something in for a, but the problem is actually asking something about b)
how would you represent:
Lindsay is twice as old as she was 10 yrs ago.
don't introduce two variables.
Let x be age now, set it equal to the alteration. For example,
Lindsay is twice as old as she was 10 yrs ago.
x= (x-10)*2
Any side of a triangle must be greater than..
the difference of the two other sides.
For example, If two sides of a triangle are 11 & 7, then 11-7=4
the other side must be greater than four
(2*10)^10=
either
20^(10)
or
(2^(10)*10^(10)
For data analysis questions, if you are asked which year has increased the most relative to the previous year..
check the steepness of the line. The line with the largest slope will have increased the most.
when comparing fractions, a larger denominator
generally means a smaller number

ie, 2/3 is smaller than 1/2

unless the numerator is lg than the denom, as is the case in the following example:

4/2 = 2
15/5=3
8*1 =
8

one times any number is the other number
sum means
product means
product means multiply
What is the volume of a cube whose total surface area is 54?
SA = (area of one face)*(number of faces)

54 = (area of one face)*(6 faces in a cube)
9 = area of one face
9 = side ^(2)
3 = side

Volume = l *W * h
volume = (side)*(side) *(side)
volume = 27
replace variables with numbers when...
-the five choices use the variable mentioned in the question.

-use a number that is easy to work with
How would you express the area of a circle in terms of C, its circumference
C= 2pi r
C/2pi = r

A = pi r ^(2)
A = pi (c/2 pi)*(c/2 pi)
A = C^(2)/4 pi

treat pi like it is x. x^2/x = x
What is the sum of integers from 1 to 50?
n(n+1)/2
50(51)/2
10^5 = (5*2)^5
(5*2)^5 can be further broken down into...
5^(5) *2^(5)
important techniques
-scratch carried numbers when multiplying a multi-step decimal prob
-use decimals instead of fract if it looks like numb won't factor easily
-when crossing out factors, circle what remains to be multiplied
-in probs that have variables in the question and in the answers, plug in values. if more than one ans fits, plug in a second set of numbers
-in quant comparison ask first, could they be equal? and then, must they be equal?
(note: the two sides will always be somewhat close together, so if they aren't I should check to see if I've made a math error )
-D is usually the strongest possibility when a range of numbers are given, not when there is a discrete value for each number.
-remember:
side 3 < than side 1+2
side 3 > side 1-2
how would you represent the following:

the sum of all the integers from 19 to 59 inclusive.
(a+b) *(b/2)
(19+59) *(59/2)

(If adding from some number to another number, the number that you're adding up to - the larger number- should be used as b)
what % of 120 is 10?

how do you represent what %
what % is represented by x/100, with which you should solve for x.
Also, note that the "of" 120 means that you should MULTIPLY 120 by x/100.
In general, when there are 2 unknowns and only one equation...
there is not enough information to make a unique solution.
When there are 2 unknowns and 2 equations...
there is usually enough information to find a unique solution. But here's something very important to watch out for. This only works if the second equation gives new information.
Two differences between inequalities and equations:
1. when you multiply by a negative number you have to switch the sign
2. when you take the inverse, or reciprocal, of each side, you also reverse the sign
Joan has twice as many dogs as cats. If she had four more cats, she would have the same number of cats as dogs. How many dogs does she have?
2c=d
c+4 =d
she has 8 dogs
Properties of parallelograms
1. opposite sides are equal
2. opposite angles are equal
3. consecutive angles (angles next to each other) add up to 180)
adding or subtracting the same expression to both sides of an inequality ...
does not change the direction of the inequality.

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